is introduced, where \(\Delta \Psi \)  is the angular separation between GRB and neutrino, and \(\sigma =\sqrt{\sigma _{\rm GRB}^2+\sigma _{\nu }^2}\)  is the standard deviation based on the angular uncertainties of GRB and neutrino measurements. The detailed values of both \(\Delta \Psi \)  and \(\sigma \)  can be found in the previous studies [1]}, [2]}, [3]}. In our analysis, we consider the GRBs whose angular separation is smaller than \(3\sigma \) .
For the redshift, some of the GRB candidates do not have determined redshift yet. As discussed in previous studies [1]}, [2]}, [3]}, here we use a most likely estimated value obtained as the average value of all known redshifts that have been measured so far. What needs to be emphasized is that a relatively wide error range of \(0.5z\sim 2z\)  is considered for the unknown redshifts in our analysis. Such a range covers most of the known redshifts of GRBs detected so far and can also separate the “long burst" and the “short burst", since the two error ranges have no overlap. In our analysis, the error of redshift plays a role in the errors of fitting parameters.
Among the 24 “shower” events above 60 TeV, 12 events can be associated with GRBs. The properties of probable GRB neutrinos that satisfy both time and direction criteria are listed in Table REF .
